Postprint. This is the accepted version of a paper presented at Nordic Concrete Research.

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1 Postprint This is the accepted version of a paper presented at Nordic Concrete Research. Citation for the original published paper: Gasch, T., Ansell, A. (2017) Influence of varying ambient conditions on time-dependent deformations inconcrete using multi-field modelling. In: Marianne Tange Holst (ed.), XXIII Nordic Concrete Research Meeting Nordic Concrete Research Publications N.B. When citing this work, cite the original published paper. Permanent link to this version:

2 1 Influence of varying ambient conditions on time-dependent deformations in concrete using multi-field modelling Tobias Gasch Tech. Lic, Ph.D. student KTH Royal Institute of Technology Division of Concrete Structures SE STOCKHOLM, Sweden Anders Ansell Ph.D., Professor KTH Royal Institute of Technology Division of Concrete Structures SE STOCKHOLM, Sweden The table lines will be removed by the editor! ABSTRACT Time-dependent deformations, such as creep and shrinkage, are important when dealing with durability aspects of concrete. In the current study, a multi-field analysis method is described, verified and used in a numerical study to investigate the influence of short and long term variations in temperature and relative humidity. It is found that especially the creep behaviour is significantly influenced by the seasonal variations in climate conditions and also to a lesser extent the daily variations. Key words: Cracking, Creep, Modelling, Multi-field, Shrinkage 1. INTRODUCTION Time-dependent deformations are important to consider when studying durability of concrete structures, and effects such as creep and shrinkage have proven difficult to explain in theory and to model using continuum based approaches. In this study, a numerical investigation is made on the effect of varying environmental conditions on time-dependent deformations in mature concrete. 2. MATHEMATICAL MODEL Although the main physical property of interest here is deformations, the time-dependent behaviour of concrete is necessarily a multi-physical problem. At a minimum the temperature and humidity distributions inside the material should be considered. With this in mind, a multifield model for the analysis of concrete including effects such as creep, shrinkage and cracking was presented by the authors in [1-2] and used for this study. A brief overview of the most important features of the model is given in the following. 2.1 Governing equations of the multi-field problem The deformations are governed by the quasi-static form of the linear momentum balance in Eq. (1) where the stress balances the self-weight ρ. Also in Eq. (1), the used definition of strain can be seen, which corresponds to an assumption of small displacements.

3 2 0 with 1 2 (1) The distribution of the temperature follows the heat equation given in Eq. (2), where heat flux is governed by Fourier s law. Parameters and are the heat capacity and thermal conductivity, respectively. 0 (2) Lastly, the distribution of moisture is given by Eq. (3), with the relative humidity as the driving potential of moisture flux. 0 (3) The flux is controlled by the non-linear transport coefficient and the sorption isotherm, which slope is described by /. Following the model of Bazant and Najjar [3], / is here assumed constant and combined with the transport coefficient, which then directly relates the flux to changes in humidity. 2.2 Boundary conditions To complete the model, the previously defined equations need to be complemented with appropriate boundary conditions. For the mechanical part, both prescribed displacements and fluxes (loads) are used, while for the heat and moisture parts also convective type boundary conditions are used. 2.3 Constitutive behaviour for time-dependent deformations The constitutive equation needed in Eq. (1) to relate and used in [1-2] to describe the timedependent deformation in concrete is given as: 1 :,, (4) The different features of Eq. (4) are described briefly in the following. Instantaneous and elastic deformation is governed by the elasticity tensor defined using a non-aging elastic modulus and Poisson s ratio. Aging of the elastic behaviour is included in the creep model [4]. Creep, as well as aging, is included using the Microprestress-Solidification (MPS) theory developed by Bazant and co-workers [4]. Creep in the MPS theory is described in part by solidifying nonaging viscelasticty (, ) accounting for short-term creep and an aging viscous part (, ) in which effects such as drying creep is accounted for. Temperature and humidity effects on aging are included using an equivalent time. Here and in [1-2], the version of the MPS theory proposed by Jirasek and Havlsek [5] is used. Drying shrinkage ( ) and thermal expansion ( ) is assumed linearly dependent on changes in and, respectively. Although not the focus of this study, cracking is important for time-dependent deformations, especially drying shrinkage. Its effect is here accounted for using a scalar isotropic damage model [6], where cracking is described by the damage parameter that depends explicitly on.

4 3 3. NUMERICAL IMPLEMENTATION Numerical implementation of the governing equations (1-3) of the model is made using standard procedures of finite element analysis and solved with the software Comsol Multiphysics [7]. For the constitutive equation (4), all parts related to creep are integrated in time using an exponential algorithm [4] in a separate subroutine. Implementation of the damage model is described in [6]. 4. APPLICATION OF THE MODEL The multi-field model is first calibrated and verified using laboratory measurements followed by a numerical example using the same test set-up but with in-situ measured ambient conditions. 4.1 Comparison with laboratory testing The model is calibrated using the experimental series by Bryant and Vadhanavikkit [8]; in particular the series using prisms with dimensions mm 3. Loaded prisms are subjected to an axial compressive load of 7 MPa and subjected to two-sided drying to simulate a continuous wall/slab with an ambient relative humidity of 0.6. First, however, total strains for fully sealed prisms are shown in Figure 1 (left) to verify the basic creep part of the model. Next, in Figure 1 (right), total strains for prisms subjected to both loading and drying are shown together with an unloaded reference case. More information on material properties and the model parameters used can be found in [1] and additional verification examples in [2]. Figure 1 Basic creep (left) and drying creep (right) for specimens loaded at different times. Solid lines show simulations and markers experimental data by Bryant and Vadhanavikkit [8]. 4.2 Effect of variable temperature and humidity under real weather conditions The calibrated model is next used to study the influence of varying ambient conditions (temperature and relative humidity) on the time-dependent deformations of concrete. The same test-specimens and load as in the laboratory setting is used, but only prisms loaded at an equivalent age of 28 days is considered. To obtain an in-situ estimate of the ambient conditions, climate data is taken from the ASHRAE Weather Data Viewer 5.0 as included in Comsol [7] for the metrological station at Flyvestation Aalborg in Hourly variation of relative humidity and temperature are shown in Figure 2 (left) together with annual means (bold solid lines). Three test cases are investigated: concrete casted in either January or July and a reference case using the annual means of relative humidity and temperature. The total deformation for all three cases is shown in Figure 2 (right). It can be clearly seen that there is a difference in the three, although after 1 year of sustained loading the specimens subjected to hourly variation have reached a similar magnitude of deformation. However, the reference case exhibits a significantly

5 4 lower deformation. Apart from seasonal thermal effects, this is mainly due to the influence that varying ambient conditions has on aging and especially the viscous strain component,. It can be concluded that it is mainly the seasonal variation that plays an important role, but also daily variations has an effect on the aging behaviour. Figure 2 Left: Hourly variation of temperature and humidity in Aalborg Right: Deformations under different ambient weather conditions. 5. CONCLUSIONS A multi-field model for time-dependent deformations in concrete (cracking, creep, shrinkage and thermal expansion) has been implemented and verified using experimental data. In a numerical investigation it was, furthermore, shown that ambient climate conditions, especially their seasonal variation, have a significant influence on the total deformations. ACKNOWLEDGMENT The research presented was carried out as a part of Swedish Hydropower Centre -SVC. SVC has been established by the Swedish Energy Agency, Energiforsk and Svenska Kraftnät together with Luleå University of Technology, KTH Royal Institute of Technology, Chalmers University of Technology and Uppsala University. REFERENCES [1] Gasch, T.: Concrete as a multi-physical material with applications to hydro power facilities, Lic. thesis, KTH Royal Institute of Technology, Stockholm, Sweden (2016) [2] Gasch T., Malm R., Ansell A.: A coupled hygro-thermo-mechanical model for concrete subjected to variable environmental conditions, Int J Solids Struct, 91: (2016) [3] Bazant Z-P., Najjar L.: Nonlinear water diffusion in nonsaturated concrete, Matriaux et Construction, 5 (1):3 20 (1972) [4] Bazant Z-P., Hauggaard A., Baweja S., Ulm F-J.: Microprestress-Solidification theory for concrete creep.i: Aging and drying effects, J Eng Mech-ASCE, 123(11): (1997) [5] Jirasek M., Havlasek P.: Microprestress-Solidification theory of concrete creep: Reformulation and improvement, Cement Concrete Res, 60(0):51 62 (2014) [6] Gasch T., Ansell A.: Cracking in quasi-brittle materials using isotropic damage mechanics, Proceedings, Comsol Conference 2016, Munich, Germany (2016) [7] Comsol Multiphysics ver 5.2a Documentation, Comsol AB, Stockholm, Sweden (2016) [8] Bryant H-A., Vadhanavikkit C.: Creep, shrinkage-size, and age at loading effects, ACI Mater J, 84: (1987)